# Questions tagged [metric-embeddings]

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### Is the face lattice of the cube a polytope graph?

The face lattice of a convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
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1 vote
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### Alexandrov's uniqueness theorem in Minkowski spacetime

Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$. Each face of $P$ comes with induced metric tensor, if the face is space-like, then it is euclidean metric; every time-like face is isometric ...
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### Embedding a countably infinite metric space in $\ell^2(\mathbb Z_+)$

Suppose $(X,d)$ is a countably infinite set endowed with a metric $d$ that satisfies the following condition: Every finite subset of $X$ with the induced metric is isometric to a subset of some ...
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### Can the sequence of complete graphs coarsely embed into Hilbert space?

Basically the title. If I have the metric space which is the disjoint union of the sequence of complete graphs, and the usual graph metric, has it been shown that the metric space can be coarsely ...
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### Is there a bi-Hölder Weierstrass-type embedding of the circle into some Euclidean space?

We say that $\Phi\colon S^1\to \mathbb{R}^d$ is an $\alpha$-bi-Hölder embedding if there are constants $c_1,c_2>0$, such that $$c_1\leq \frac{\|\Phi(x)-\Phi(y)\|}{d(x,y)^\alpha}\leq c_2,$$ where $d$...
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### Are hyperbolic spaces actually better for embedding trees than Euclidean spaces?

There is a folklore in the empirical computer-science literature that, given a tree $(X,d)$, one can find a bi-Lipschitz embedding into a hyperbolic space $\mathbb{H}^n$ and that $n$ is "much ...
1 vote
95 views

### Best estimate on doubling constant of a finite metric space

Let $(X,d)$ be a finite metric space. Clearly, $(X,d)$ is a doubling metric space but is there a 'best' estimate of $(X,d)$'s doubling constant? Probability based on its cardinality, diameter, and ...
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### Does there exist a countable metric space which is Lipschitz universal for all countable metric spaces?

Is there a countable metric space $U$ such that any countable metric space is bi-Lipschitz equivalent to a subset of $U$? How about $c_{00}(\mathbb{Q})$ where $\mathbb{Q}$ is the rational numbers? ...
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1 vote
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### intuition about Gaussian processes over a finite space

In a paper that I am reading the authors defines $\mathbb P(n,q)$ the space of covariance tensors for $\mathbb R^q$-valued Gaussian processes on an abstract finite space $K=\{x_1,\dots,x_n\}$. In his ...
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### Kernels with finite dimensional feature spaces

Suppose $x,y \in \mathbb{R}^n$ for some given fixed n. Consider a kernel $K(x,y) = f(\langle x, y \rangle)$, I'd like to know which functions $f$ admit a finite dimensional feature map. In other words,...
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### When is a metric space a snowflake?

Let $(X,d)$ be a metric space. For any $0<\epsilon<1$, we call the metric space $(X,d^{\epsilon})$; where $d^{\epsilon}(x,y)\triangleq (d(x,y))^{\epsilon}$ the $\epsilon$-snowflake of $(X,d)$. ...
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### Why $(\mathrm{Lip}([0,1]^2))^*$ is finitely representable in 1-Wasserstein space over the plane?

In "Snowflake universality of Wasserstein spaces"" by Alexandr Andoni, Assaf Naor, and Ofer Neiman, they have the following notation: For a metric space X they write $\mathcal{P}_1(X)$ ...
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### Bi-Hölder embeddings of finite metric spaces

This is a reference request. There is a large body of work, I'm familiar with, that describes the existence of bi-Hölder embeddings of finite metric spaces into Euclidean space (such as this ...
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### Correspondence between Riemannian metrics and Euclidean embeddings

Given a sufficiently smooth manifold M, a Riemannian metric on M induces an isometric embedding into Euclidean space by Nash's theorem, (non-canonically, non-uniquely) an embedding of M into ...
137 views

### Banach embedding of finite dimensional spaces

Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]). I am wondering whether are any result for the case when $r>s>2$? ...
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### An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1

The following question is related to this previous question, Canonical immersion of the double torus: Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ...
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### Dense embeddings into Euclidean space

The question is a follow-up on this old post. Fix a positive integer $d$ and consider $\mathbb{R}^d$ with its usual Euclidean topology. Given a metric space $(X,\delta_X)$, what conditions are ...
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### Two questions around the $abc$-conjecture

Let $d(a,b) = 1-\frac{2 \gcd(a,b)}{a+b}$, $d_{ABC}(a,b) = 1-\frac{2\gcd(a,b)^3}{ab(a+b)}$ be two metrics on natural numbers. The abc-conjecture can be formulated using these two metrics as: For ...
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### Is the matrix $\mu_f(X_i \cap X_j)$ positive definite?

Let $X_1,\ldots, X_n$ be finite subsets of some larger finite set $Z$. Let $f:Z \rightarrow \mathbb{R}_{>0}$ be any function, and define a (counting) measure $\mu_f(X) = \sum_{x \in X} f(x)$ for a ...
1 vote
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